Theorem elxp6 8069

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7968. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
elxp6	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Proof of Theorem elxp6

Step	Hyp	Ref	Expression
1		elxp4 7968	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
2		1stval 8037	. . . . 5 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}
3		2ndval 8038	. . . . 5 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
4	2, 3	opeq12i 4850	. . . 4 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩
5	4	eqeq2i 2776	. . 3 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ↔ 𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩)
6	2	eleq1i 2855	. . . 4 ⊢ ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵)
7	3	eleq1i 2855	. . . 4 ⊢ ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶)
8	6, 7	anbi12i 640	. . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))
9	5, 8	anbi12i 640	. 2 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
10	1, 9	bitr4i 281	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {csn 4588  ⟨cop 4840  ∪ cuni 4872   × cxp 5656  dom cdm 5658  ran crn 5659  ‘cfv 6566  1^st c1st 8033  2^nd c2nd 8034

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-1st 8035  df-2nd 8036

This theorem is used by:  elxp7  8070  eqopi  8071  1st2nd2  8074  eldju2ndl  10042  eldju2ndr  10043  r0weon  10130  qredeu  16877  qnumdencl  16959  setsstruct2  17396  tx1cn  23983  tx2cn  23984  txhaus  24021  psmetxrge0  24687  xppreima  33337  ofpreima2  33358  smatrcl  34539  1stmbfm  35003  2ndmbfm  35004  oddpwdcv  35098  prproropf1olem0  48857